First equation on page

Eq. 11.42

Text following Eq. 11.42

Eq. 11.43

Eq. 11.44

Eq. 11.45

Text following Eq. 11.49

Comparing the last result with Eq. 11.10, we see that the curvature
given by
is constant everywhere and that the principal normal
vector at any point on the helix is pointed toward the helix axis and is perpendicular
to it. Also, the vectors and are constant vectors independent of
. If we now take the axis of the helix to be along
so that
and
put the results above into Eq. 11.46, we find that it is satisfied if

Eq. 18.49

Eq. 18.57

Eq. 18.58

Eq. 18.60

Fig. 19.10

Eq. 19.82

Eq. 19.83

Eq. 19.84

Text before Eq. 19.86

Standard relationships for volumes and areas show that the volume
of the nucleus is given by
while the area of the two facets is
and the area of the spherical portion of the interface is
. The free energy to form the nucleus is therefore

Eq. 19.86

Eq. 19.87

Eq. 19.88

Eq. 19.101

Eq. 19.102

Eq. 19.103

Eq. 19.104

Eq. 19.106

Eq. 19.108

text before Eq. 19.115

Let
,
, and
be the energies (per unit area) of the liquid/particle,
liquid/solid, and solid/particle interfaces, respectively. From Section 19.2.1
the volume of the solid nucleus is
, the spherical liquid/solid cap area is
, and the solid/particle area
is
. The free energy of nucleus formation
on the particles is then

Eq. 19.117

text after 19.117

On the other hand, for
the homogeneous nucleation of a spherical solid nucleus in the bulk liquid,
.

Eq. 20.81

Eq. 20.82

Eq. 20.83

Eq. 20.84

Eq. 20.85

Eq. 20.88

Eq. 20.89

Eq. 20.91

Eq. 20.96

Eq. 20.97

Eq. 20.98

Eq. 20.101

Eq. 20.102

Eq. 20.103

Text after 20.103

Using the linear approximation
in Fig. 20.14,
. Also, the conservation
of atoms requires that the two shaded areas in the figure be equal.
Therefore,
. Putting these relationships
into Eq. 20.103 gives